We investigate long-range coherent and dissipative coupling between two spatially separated magnets while both are coupled to a microwave cavity. A careful examination of the system shows that the indirect interaction between two magnon modes is dependent on their individual mechanisms of direct coupling to the cavity. If both magnon modes share the same form of coupling to the cavity (either coherent or dissipative), then the indirect coupling between them will produce level repulsion. Conversely, if the magnon modes have different forms of coupling to the cavity (one coherent and one dissipative), then their indirect coupling will produce level attraction. We further demonstrate the cavity-mediate nature of the indirect interaction through investigating the dependence of the indirect coupling strength on the frequency detuning between the magnon and cavity modes. Our work theoretically and experimentally explores indirect cavity mediate interactions in systems exhibiting both coherent and dissipative coupling, which opens a new avenue for controlling and utilizing light-matter interactions.
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